The value of `((1)/(2)" of "1(1)/(2))-:(3(1)/(2)-1(1)/(4))" of "1(1)/(2)-1(1)/(2)-:2(1)/(4)+1(1)/(3)` is:

Options

A. `(2)/(9)`
B. `(8)/(9)`
C. `(2)/(3)`
D. `(4)/(3)`

Answer

To solve the expression given in the question, we will break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert all the mixed numbers into improper fractions for easier calculations.

- \(1 \frac{1}{2} = \frac{3}{2}\)
- \(3 \frac{1}{2} = \frac{7}{2}\)
- \(1 \frac{1}{4} = \frac{5}{4}\)
- \(2 \frac{1}{4} = \frac{9}{4}\)
- \(1 \frac{1}{3} = \frac{4}{3}\)

### Step 2: Rewrite the Expression
Now we can rewrite the expression using the improper fractions:

\[ \frac{1}{2} \cdot \frac{3}{2} \div \left(\frac{7}{2} - \frac{5}{4}\right) \cdot \left(\frac{3}{2} - \frac{3}{2}\right) \div \left(\frac{9}{4} + \frac{4}{3}\right) \]

### Step 3: Calculate the Denominator
Calculate \( \frac{7}{2} - \frac{5}{4} \):

To subtract these fractions, we need a common denominator. The least common multiple of 2 and 4 is 4.

\[ \frac{7}{2} = \frac{14}{4} \quad \text{(multiply numerator and denominator by 2)} \]
\[ \frac{14}{4} - \frac{5}{4} = \frac{14 - 5}{4} = \frac{9}{4} \]

### Step 4: Calculate the Second Denominator
Now calculate \( \frac{3}{2} - \frac{3}{2} \):

\[ \frac{3}{2} - \frac{3}{2} = 0 \]

### Step 5: Calculate the Third Denominator
Calculate \( \frac{9}{4} + \frac{4}{3} \):

To add these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12.

\[ \frac{9}{4} = \frac{27}{12} \quad \text{(multiply numerator and denominator by 3)} \]
\[ \frac{4}{3} = \frac{16}{12} \quad \text{(multiply numerator and denominator by 4)} \]
\[ \frac{27}{12} + \frac{16}{12} = \frac{43}{12} \]

### Step 6: Substitute Back into the Expression
Now substitute back into the expression:

\[ \frac{1}{2} \cdot \frac{3}{2} \div \left(\frac{9}{4}\right) \cdot 0 \div \left(\frac{43}{12}\right) \]

### Step 7: Simplify the Expression
Since we have a multiplication by zero:

\[ \text{The entire expression evaluates to } 0. \]

### Final Answer
The value of the expression is \(0\).

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